Take the kinetic energy operator and ground state wave function for a particle in a box with width of $L$ as an example (in 1D)
$$\hat{T}=-\frac{\hbar}{2m}\frac{d}{dx^2}$$
and
$$\psi=\sqrt{\frac{2}{L}}\sin\frac{\pi}{L}x$$
Obviously, $\hat{T}\psi$ means something and $\psi^{*}\hat{T}$ means something else.